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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034. M.Sc. DEGREE EXAMINATION PHYSICS SECOND SEMESTER APRIL 2003 PH 2803 / PH 825 MATHEMATICAL PHYSICS 28.04.2003 1.00 4.00 Max : 100 Marks PART A (10 2=20 marks) Answer ALL questions. 01. Starting from the general equation of a circle in the xy plane A (x2 +y2) + Bx + Cy +D=0 arrive at the z z* representation for a circle. 02. State Liouville’s theorem. z 03. Develop Laurent series of f ( z ) about z = -2. ( z 1)( z 2) 04. Write the Jacobian of the transformation w 2 z 2 i z 3 i . 1 05. Show that the Dirac delta function (ax) ( x) for a 0 . a 06. State convolution theorem. 07. Solve the differential equation T + k 2 T 0 . x 2x ...... in the interval 08. Obtain the orthonormalising constant for the series cos , cos L L (L, L). 09. Evaluate 4 x2 3 d x using the knowledge of Gamma function. 0 10. Generate L2 (x) and L3 (x) using Rodrigue’s formula for laugerre polynomials. PART B (4 7.5=30 marks) Answer any FOUR. 11. Obtain Cauchy Rieman equations from first principles of calculus of complex numbers. 12. Determine a function which maps the indicated region of w plane on to the upper half of the z plane v y w plane z plane p T Q s u p1 Q1 S1 T1 -b +b -1 +1 13. Develop halfrange Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the (1) n 1 2 results to develop the series . 12 n 1 n 2 X 14. Verify that the system y11 + y 0 ; y1(0) = 0 and y (1) = 0 is a SturmLiouville System. Find the eigen values and eigen functions of the system and hence form a orthnormal set of functions. 1 2 2A 15. (a) If f (x) = Ak Pk ( x) obtain Parseval’s Identity { f ( x)} d x k k 0 k 0 2 k 1 1 where Pk (x) stands for Legendre polynomials. / (b) Prove that H n (x) = 2n - 1 Hn (x) where Hn (x) stands for Hermite polynomials.(4+3.5) 2 PART C (412.5=50 marks) Answer any FOUR. 16. Show that u (x, y) = Sin x Coshy + 2 Cos x Sinhy + x2 +4 xy y2 is harmonic Construct f (z) such that u + iv is analytic. 2 d 17. (a) Evaluate using contour integration. 5 4 sin 0 ez d z (b) Using suitable theorems evaluate 2 c : z i 2. (7+5.5) c z 4 18. (a) The current i and the charge q in a series circuit containing an inductance L and dq di q E and i = capacitance C and emf E satisfy the equations L . Using dt c dt Laplace Transforms solve the equation and express i interms of circuit parameters. 1 (b) Find L1 2 , where L-1 stands for inverse Laplace transform. (3.5) 2 2 (s a ) 19. Solve the boundary value problem 2 y 2 y 2 a . with Y (0, t) = 0; yx (L, t) = 0 t2 x2 y (x, 0) = f (x) ; yt (x, 0) = 0 and y ( x, t ) M . and Interpret physically. 20. Solve Bessels differential equation using Froebenius power series method. + + + + +